Asymptotic stability and boundedness for functional differential equations

We study a system(D)x′=F(t,x _t) of functional differential equations, together with a scalar equation(S)x′=−a(t)f(x)+b(t)g(x(t−h)) as well as perturbed forms. A Liapunov functional is constructed which has a derivative of a nature that has been widely discussed in the literature. On the basis of this example we prove results for (D) on asymptotic stability and equi-boundedness. Supported in part by NSF of China, Key Project # 19331060     

On the estimation of solutions for some linear systems of differential equations

In this paper we estimate the solutions of homogeneous linear system of differential equations with unbounded coefficients on the real lineR. We also give a necessary and sufficient condition in order that the linear differential operator with unbounded coefficients has a bounded inverse in the scalar case.     

Limiting equations in asymptotic stability problems for nonautonomous differential equations

Summary By limiting equations, we prove some asymptotic stability theorems for the origin ofR ⁿ with respect to the solutions of a differential equation , also when the functionf is not defined forx=0. Further we examine similar problems concerning the asymptotic stability of a setM ofR ⁿ that can be unbounded.

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Riassunto Mediante le equazioni limiti, si dimostrano alcuni teoremi di stabilità asintotica per l'origine diR ⁿ rispetto alle soluzioni di un'equazione differenziale , anche quando la funzionef non è definita perx=0. Vengono inoltre esaminati analoghi problemi relativi alla stabilità asintotica di un insiemeM diR ⁿ anche non limitato.

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Work performed under the auspices of the Italian Council of Research (G.N.F.M. del C.N.R.).     

Critical points at infinity and blow up of solutions of autonomous polynomial differential systems via compactification

Critical points at infinity for autonomous differential systems are defined and used as an essential tool. Rn is mapped onto the unit ball by various mappings and the boundary points of the ball are used to distinguish between different directions at infinity. These mappings are special cases of compactifications. It is proved that the definition of the critical points at infinity is independent of the choice of the mapping to the unit ball.We study the rate of blow up of solutions in autonomous polynomial differential systems of equations via compactification methods. To this end we represent each solution as a quotient of a vector valued function (which is a solution of an associated autonomous system) by a scalar function (which is a solution of a related scalar equation).     

Oscillation of nonlinear vector differential equations

Summary Oscillation criteria are obtained for vector partial differential equations of the type Δv+b(x, v)v=0, x∈G, v∈E^m, where G is an exterior domain in E_n, and b is a continuous nonnegative valued function in G × E^m. A solution v: G→E^m is called h-oscillatory in G whenever the scalar product [v(x), h] (|h|=1) has zeros x in G with |x| arbitrarily large. It is shown that the spherical mean of [v(x), h] over a hypersphere of radius r in Eⁿ satisfies a nonlinear ordinary differential inequality. As a consequence, the main theorems give sufficient conditions on b(x, t), depending upon the dimension n, for all solutions v to be h-oscillatory in G. Entrata in Redazione il 26 giugno 1975.     

Stability of Stochastic Delay Differential Equation with a Small Parameter

Abstract

Stochastic delay differential equations with wideband noise perturbations is considered. First it is shown that the perturbed system converges weakly to a stochastic delay differential equation driven by a Brownian motion. Stability and asymptotic properties of stochastic delay differential equations with a small parameter are developed. It is shown that the properties such as stability, recurrence, etc., of the limit system with time lag is preserved for the solution x ^?(·) of the underlying delay equation for ? > 0 small enough. Perturbed Liapunov function method is used in the analysis.     

Topological conditions for the existence of bounded solutions of quasihomogeneous problems

We consider a system of differential equations =P(x,t) + X(x,t), (x,t) Rⁿ* R where P C⁷(Rⁿ* R) and is a positively homogeneous function of x of degree m, larger than one, and the function X is small in comparison with P at infinity. In terms of the Lyapunov-Krasovskii function of the corresponding homogeneous system a certain submanifold of the unit sphere is defined. It is shown that if this submanifold is not contractible, then the quasihomogeneous system being considered has at least one bounded solution. The proof is based on the topological principle of Wazewski.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 143, pp. 162–169, 1985.     

The cauchy problem of a degenerate parabolic equation

This note is concerned with the existence of a weak solution for a degenerate Cauchy problem of parabolic type in then-dimensional spaceR ⁿ. The degenerate property is in the sense that the matrix (a _ij(t,x)) involved in the differential operator is not necessarily positive definite. The essential idea is the construction of a suitable function spaceH and to prove the existence of a weak solution inH.     

Isometric approximation property of unbounded sets

We give a necessary and sufficient quantitative geometric condition for an unbounded set A ? Rⁿ to have the following property with a given c > 0: For every ε ≥ 0 and for every map f: A → Rⁿ such that , there is an isometry T: A → Rⁿ such that ¦Tx?fx¦ ≤ cε for all x ∈ A.     

The design centering problem as a D.C. programming problem

The following problem is studied: Given a compact setS inR ⁿ and a Minkowski functionalp(x), find the largest positive numberr for which there existsx S such that the set of ally R ⁿ satisfyingp(y–x) r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this design centering problem can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR ⁿ . In the case where, moreover,p(x) = (x ^T Ax)^1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.     

On Birkhoff interpolation (0;2) case

P. Turán and his associates^[2] considered in detail the problem of (0,2) interpolation based on the zeros of π_n(x). Motivated by these results and an earlier result of Szabados and Varma^[9] here we consider the problem of existence, uniqueness and explicit representation of the interpolatory polynomial R_n(x) satis fying the function values at one set of nodes and the second derivative on the other set of nodes. It is important to note that this problem has a unique solution provided these two sets of nodes are chosen properly. We also promise to have an interesting convergence theorem in the second paper of this series, which will provide a solution to the related open problem of P. Turán.     

On matrix measures and convex Liapunov functions

In this paper, we extend the concept of the measure of a matrix to encompass a measure induced by an arbitrary convex positive definite function. It is shown that this “modified” matrix measure has most of the properties of the usual matrix measure, and that many of the known applications of the usual matrix measure can therefore be carried over to the modified matrix measure. These applications include deriving conditions for a mapping to be a diffeomorphism on Rⁿ, and estimating the solution errors that result when a nonlinear network is approximated by a piecewise linear network. We also develop a connection between matrix measures and Liapunov functions. Specifically, we show that if V is a convex positive definite function and A is a Hurwitz matrix, then μ_V(A) < 0, if and only if V is a Liapunov function for the system $\text{x}\text{?}\text{= Ax}$. This linking up between matrix measures and Liapunov functions leads to some results on the existence of a “common” matrix measure μ_V(·) such that μ_V(A_i) < 0 for each of a given set of matrices A₁,…, A_m. Finally, we also give some results for matrices with nonnegative off-diagonal terms.     

Asymptotic dynamics of plate equations with a critical exponent on unbounded domain

The long-time behavior of plate equations with a critical exponent on the unbounded domain Rⁿ${\mathbb{R}}^n$ is studied. We show that there exists a compact global attractor. The attractor is characterized as the unstable manifold of the set of stationary points, due to the existence of a Lyapunov functional.     

Criteria for validity of the maximum modulus principle for solutions of linear parabolic systems

We consider systems of partial differential equations of the first order int and of order 2s in thex variables, which are uniformly parabolic in the sense of Petrovskii. We show that the classical maximum modulus principle is not valid inR ⁿ×(0,T] fors≥2. For second order systems we obtain necessary and, separately, sufficient conditions for the classical maximum modulus principle, to hold in the layerR ⁿ×(0,T] and in the cylinder μ×(0,T], where μ is a bounded subdomain ofR ⁿ. If the coefficients of the system do not depend ont, these conditions coincide. The necessary and sufficient condition in this case is that the principal part of the system is scalar and that the coefficients of the system satisfy a certain algebraic inequality. We show by an example that the scalar character of the principal part of the system everywhere in the domain is not necessary for validity of the classical maximum modulus principle when the coefficients depend both onx andt. The research of the first author was supported by the Ministry of Absorption, State of Israel.     

A necessary and sufficient condition for the nirenberg problem

We seek metrics conformal to the standard ones on Sⁿ having prescribed Gaussian curvature in case n = 2 (the Nirenberg Problem), or prescribed scalar curvature for n ≧ 3 (the Kazdan-Warner problem). There are well-known Kazdan-Warner and Bourguignon-Ezin necessary conditions for a function R(x) to be the scalar curvature of some conformally related metric. Are those necessary conditions also sufficient? This problem has been open for many years. In a previous paper, we answered the question negatively by providing a family of counter examples. In this paper, we obtain much stronger results. We show that, in all dimensions, if R(x) is rotationally symmetric and monotone in the region where it is positive, then the problem has no solution at all. It follows that, on S², for a non-degenerate, rotationally symmetric function R(θ), a necessary and sufficient condition for the problem to have a solution is that Rθ changes signs in the region where it is positive. This condition, however, is still not sufficient to guarantee the existence of a rotationally symmetric solution, as will be shown in this paper. We also consider similar necessary conditions for non-symmetric functions. ©1995 John Wiley & Sons, Inc.     

UNIFORM ASYMPTOTIC STABILITY OF FUNCTIONAL DIFFERENTIAL EQUATIONS

With the help of a Liapunov functional with seml-negative definite derivative,Barbashin-Krasovskii's theorem is extended to nonautonomous functional differentialequations,a reducing dimension approach is presented for the uniform asymptotic stabilityof high dimension systems,and some sufficient conditions of uniform asymptotic stabilityare obtained.     

Orthogonality criteria for compactly supported refinable functions and refinable function vectors

A refinable function φ(x):ℝⁿ→ℝ or, more generally, a refinable function vector Φ(x)=[φ₁(x),...,φ_r(x)]^T is an L¹ solution of a system of (vector-valued) refinement equations involving expansion by a dilation matrix A, which is an expanding integer matrix. A refinable function vector is called orthogonal if {φ_j(x−α):α∈ℤⁿ, 1≤j≤r form an orthogonal set of functions in L²(ℝⁿ). Compactly supported orthogonal refinable functions and function vectors can be used to construct orthonormal wavelet and multiwavelet bases of L²(ℝⁿ). In this paper we give a comprehensive set of necessary and sufficient conditions for the orthogonality of compactly supported refinable functions and refinable function vectors.     

Periodic boundary value problem of functional differential equations with perturbation

By coincidence degree, the existence of solution to the periodic boundary value problem of functional differential equations with perturbation     

Fractional partial differential equations and modified Riemann-Liouville derivative new methods for solution

The paper deals with the solution of some fractional partial differential equations obtained by substituting modified Riemann-Liouville derivatives for the customary derivatives. This derivative is introduced to avoid using the so-called Caputo fractional derivative which, at the extreme, says that, if you want to get the first derivative of a function you must before have at hand its second derivative. Firstly, one gives a brief background on the fractional Taylor series of nondifferentiable functions and its consequence on the derivative chain rule. Then one considers linear fractional partial differential equations with constant coefficients, and one shows how, in some instances, one can obtain their solutions on by-passing the use of Fourier transform and/or Laplace transform. Later one develops a Lagrange method via characteristics for some linear fractional differential equations with nonconstant coefficients, and involving fractional derivatives of only one order. The key is the fractional Taylor series of non differentiable functionf(x + h) =E _α (h ^α D _x ^α )f(x).     

Interaction of acoustic waves and piezoelectric structures

In the paper, we investigate the basic transmission problems arising in the model of fluid‐solid acoustic interaction when a piezo‐ceramic elastic body ( Ω ⁺ ) is embedded in an unbounded fluid domain ( Ω ^? ). The corresponding physical process is described by boundary‐transmission problems for second order partial differential equations. In particular, in the bounded domain Ω ⁺ , we have 4 × 4 dimensional matrix strongly elliptic second order partial differential equation, while in the unbounded complement domain Ω ^? , we have a scalar Helmholtz equation describing acoustic wave propagation. The physical kinematic and dynamic relations mathematically are described by appropriate boundary and transmission conditions. With the help of the potential method and theory of pseudodifferential equations, the uniqueness and existence theorems are proved in Sobolev–Slobodetskii spaces. Copyright © 2014 John Wiley & Sons, Ltd.